Do I need to own all the rights to the music I'm selling?

You must own the copyright for the sound recordings or have the authority or permission from the owner(s).

If you didn't write the song/composition, that's OK, but you must find out who the copyright owners are, and pay the publisher their mechanical royalties the same way you would for CDs sold, but based on your download/sales activity.

If you have samples in your music, they must all be legally cleared and paid-for. No "mix tapes" of other people's music, even if you are mixing in your own music.

It's VERY important you have all rights and permission! Files distributed on the internet are watched very carefully by lawyers. You can't just "get away with it". Do everything thoroughly and legit.